Percentage — Study Notes for MPESB Group 1/2/3
Overview
Percentage is one of the most fundamental and frequently tested topics in MPESB mathematics. It forms the backbone for several advanced topics including Profit and Loss, Simple and Compound Interest, Data Interpretation, and even questions on population growth or depreciation. Mastering percentage calculations is non-negotiable for scoring well.
The concept is straightforward — "percent" means "per hundred." Any fraction with denominator 100 can be expressed as a percentage. However, exam questions test your ability to apply this concept quickly under time pressure, often combining it with ratio, fractions, or successive changes.
Students must focus on three skills: lightning-fast fraction-to-percentage conversions, understanding percentage change (increase/decrease), and handling successive percentage changes without making sign errors.
Key Concepts
- Basic definition: x% means x/100. So 25% = 25/100 = 1/4 = 0.25.
- Converting fraction to percentage: Multiply the fraction by 100. For example, 3/5 = (3/5) × 100 = 60%.
- Converting percentage to fraction: Divide by 100 and simplify. For example, 37.5% = 37.5/100 = 3/8.
- Percentage of a quantity: x% of A = (x/100) × A. Always identify what "of" refers to — that's your base.
- Percentage increase/decrease: Change% = (Change / Original) × 100. The denominator is always the original value, not the new value.
- Successive percentage changes: When two changes of a% and b% occur successively, net effect = a + b + (ab/100)%. This formula handles signs automatically if you use + for increase and − for decrease.
- Reverse percentage: If after x% increase, value becomes A, then original = A × (100/(100+x)).
- Population/Depreciation formula: For n periods of r% growth: Final = Initial × (1 + r/100)ⁿ. For depreciation, use (1 − r/100)ⁿ.
Formulas / Key Facts
| Concept | Formula |
|---|---|
| x% of A | (x × A) / 100 |
| Fraction to % | Fraction × 100 |
| % Increase | [(New − Old) / Old] × 100 |
| % Decrease | [(Old − New) / Old] × 100 |
| Net successive change (a%, b%) | a + b + (ab/100) % |
| Finding original after x% increase | New × (100 / (100+x)) |
| Finding original after x% decrease | New × (100 / (100−x)) |
| Population after n years at r% growth | P × (1 + r/100)ⁿ |
| Depreciated value after n years | P × (1 − r/100)ⁿ |
Must-memorize fraction-percentage equivalents:
- 1/2 = 50%, 1/3 ≈ 33.33%, 1/4 = 25%, 1/5 = 20%
- 1/6 ≈ 16.67%, 1/8 = 12.5%, 1/10 = 10%, 1/12 ≈ 8.33%
- 2/3 ≈ 66.67%, 3/4 = 75%, 2/5 = 40%, 3/5 = 60%
Worked Examples
Example 1: Basic percentage calculation
A student scored 72 marks out of 90. What is the percentage?
Percentage = (72/90) × 100 = (4/5) × 100 = 80%
Example 2: Percentage increase
The price of rice increased from ₹40 per kg to ₹46 per kg. Find the percentage increase.
Increase = 46 − 40 = ₹6
% Increase = (6/40) × 100 = 15%
Example 3: Successive percentage changes
A shopkeeper increases price by 20% and then gives a 10% discount. What is the net percentage change in price?
Using formula: Net = a + b + (ab/100)
Here a = +20, b = −10
Net = 20 + (−10) + (20 × −10)/100 Net = 20 − 10 − 2 = 8%
The price effectively increases by 8%.
Example 4: Reverse percentage
After a 25% increase, the salary of an employee becomes ₹50,000. What was the original salary?
Original = New × (100 / (100 + 25)) Original = 50000 × (100/125) = 50000 × (4/5) = ₹40,000
Example 5: Population growth
The population of a town is 1,00,000. It increases by 10% in the first year and 20% in the second year. What is the population after 2 years?
After Year 1: 1,00,000 × (110/100) = 1,10,000
After Year 2: 1,10,000 × (120/100) = 1,32,000
Alternatively, multiply both factors: 1,00,000 × 1.1 × 1.2 = 1,32,000
Common Mistakes
- Wrong base selection: Students often calculate percentage change using the new value as base instead of the original value. Remember — for percentage change, always divide by the ORIGINAL quantity.
- Sign errors in successive changes: When applying the formula a + b + (ab/100), students forget that a decrease is negative. If price decreases by 20%, use a = −20, not +20.
- Confusing "by" and "to": "Increased by 20%" means new value = 120% of original. "Increased to 120" means the new value IS 120. Read carefully.
- Ignoring order in some problems: While the successive change formula gives the same result regardless of order, some word problems (like "find value after first change") need step-by-step calculation.
- Percentage point vs percentage change: If interest rate moves from 10% to 12%, it increased by 2 percentage points but by 20% in relative terms. Exams sometimes test this distinction.
Quick Reference
- x% of A = (x × A) / 100 — always multiply then divide by 100.
- % Change = (Difference / Original) × 100 — denominator is always ORIGINAL.
- Successive changes a% and b% → Net = a + b + (ab/100)% — use signs correctly.
- To find original after x% increase → Multiply new value by 100/(100+x).
- Memorize: 1/8 = 12.5%, 1/6 = 16.67%, 1/3 = 33.33%, 2/3 = 66.67%.
- Shortcut: 10% increase followed by 10% decrease ≠ no change; it results in 1% decrease.